Beyond the RPA on the cheap: improved correlation energies with the efficient "Radial Exchange Hole" kernel

The "ACFD-RPA" correlation energy functional has been widely applied to a variety of systems to successfully predict energy differences, and less successfully predict absolute correlation energies. Here we present a parameter-free exchange-correlation kernel that systematically improves absolute correlation energies, while maintaining most of the good numerical properties that make the ACFD-RPA numerically tractable. The "RXH" kernel is constructed to approximate the true exchange kernel via a carefully weighted, easily computable radial averaging. Correlation energy errors of atoms with two to eighteen electrons show a thirteenfold improvement over the RPA and a threefold improvement over the related "PGG" kernel, for a mean absolute error of 13mHa or 5%. The average error is small compared to all but the most difficult to evaluate kernels. van der Waals $C_6$ coefficients are less well predicted, but still show improvements on the RPA, especially for highly polarisable Li and Na

Efficient, long-range correlation from occupied wavefunctions only

We use continuum mechanics [Tao \emph{et al}, PRL{\bf 103},086401] to approximate the dynamic density response of interacting many-electron systems. Thence we develop a numerically efficient exchange-correlation energy functional based on the Random Phase Approximation (dRPA). The resulting binding energy curve $E(D)$ for thin parallel metal slabs at separation $D$ better agrees with full dRPA calculations than does the Local Density Approximation. We also reproduce the correct non-retarded van der Waals (vdW) power law E(D)\aeq -C_{5/2}D^{-5/2} as $D\to\infty$, unlike most vdW functionals.Comment: 4 pages, 1 figur

The flexible nature of exchange, correlation and Hartree physics: resolving "delocalization" errors in a 'correlation free' density functional

By exploiting freedoms in the definitions of 'correlation', 'exchange' and 'Hartree' physics in ensemble systems we better generalise the notion of 'exact exchange' (EXX) to systems with fractional occupations functions of the frontier orbitals, arising in the dissociation limit of some molecules. We introduce the Linear EXX ("LEXX") theory whose pair distribution and energy are explicitly \emph{piecewise linear} in the occupations $f^{\sigma}_{i}$. {\hi}We provide explicit expressions for these functions for frontier $s$ and $p$ shells. Used in an optimised effective potential (OEP) approach it yields energies bounded by the piecewise linear 'ensemble EXX' (EEXX) energy and standard fractional optimised EXX energy: $E^{EEXX}\leq E^{LEXX} \leq E^{EXX}$. Analysis of the LEXX explains the success of standard OEP methods for diatoms at large spacing, and why they can fail when both spins are allowed to be non-integer so that "ghost" Hartree interactions appear between \emph{opposite} spin electrons in the usual formula. The energy $E^{LEXX}$ contains a cancellation term for the spin ghost case. It is evaluated for H, Li and Na fractional ions with clear derivative discontinuities for all cases. The $p$-shell form reproduces accurate correlation-free energies of B-F and Al-Cl. We further test LEXX plus correlation energy calculations on fractional ions of C and F and again shows both derivative discontinuities and good agreement with exact results

Dispersion corrections in graphenic systems: a simple and effective model of binding

We combine high-level theoretical and \emph{ab initio} understanding of graphite to develop a simple, parametrised force-field model of interlayer binding in graphite, including the difficult non-pairwise-additive coupled-fluctuation dispersion interactions. The model is given as a simple additive correction to standard density functional theory (DFT) calculations, of form $\Delta U(D)=f(D)[U^{vdW}(D)-U^{DFT}(D)]$ where $D$ is the interlayer distance. The functions are parametrised by matching contact properties, and long-range dispersion to known values, and the model is found to accurately match high-level \emph{ab initio} results for graphite across a wide range of $D$ values. We employ the correction on the difficult bigraphene binding and graphite exfoliation problems, as well as lithium intercalated graphite LiC$_6$. We predict the binding energy of bigraphene to be 0.27 J/m^2, and the exfoliation energy of graphite to be 0.31 J/m^2, respectively slightly less and slightly more than the bulk layer binding energy 0.295 J/m^2/layer. Material properties of LiC$_6$ are found to be essentially unchanged compared to the local density approximation. This is appropriate in view of the relative unimportance of dispersion interactions for LiC$_6$ layer binding

A step toward density benchmarking -- the energy-relevant "mean field error"

Since the development of generalized gradient approximations in the 1990s, approximations based on density functional theory have dominated electronic structure theory calculations. Modern approximations can yield energy differences that are precise enough to be predictive in many instances, as validated by large- and small-scale benchmarking efforts. However, assessing the quality of densities has been the subject of far less attention, in part because reliable error measures are difficult to define. To this end, this work introduces the mean-field error that directly assesses the quality of densities from approximations. The mean-field error is contextualised within existing frameworks of density functional error analysis and understanding, and shown to be part of the density-driven error. It is demonstrated on several illustrative examples. Its potential use in future benchmarking protocols is discussed, and some conclusions drawn